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Géotechnique 00, 1–17 [http://dx.doi.org/10.1680/geot.XX.XXXXX]

Modelling the cyclic ratcheting of sands

through memory-enhanced bounding surface plasticity

H.Y. LIU∗, J.A. ABELL†, A. DIAMBRA‡, F. PISANÒ∗

The modelling and simulation of cyclic sand ratcheting is tackled via a plasticity model formulated within

the well-known critical state, bounding surface SANISAND framework. For this purpose, a third locus –

termed ‘memory surface’ – is cast into the constitutive formulation, so as to phenomenologically capture

micro-mechanical, fabric-related processes directly relevant to the cyclic response. The predictive capability

of the model under numerous loading cycles (‘high-cyclic’ loading) is explored with focus on drained

loading conditions, and validated against experimental test results from the literature – including triaxial,

simple shear and oedometer cyclic loading. The model proves capable of reproducing the transition from

ratcheting to shakedown response, in combination with a single set of soil parameters for diﬀerent initial,

boundary and loading conditions. This work contributes to the analysis of soil-structure interaction under

high-cyclic loading events, such as those induced by environmental and/or traﬃc loads.

KEYWORDS: sands, stiﬀness, constitutive relations, plasticity, numerical modelling, oﬀshore engineering

INTRODUCTION

Predicting the cyclic response of sands is relevant to

numerous geotechnical applications, for instance in the

ﬁelds of earthquake, oﬀshore and railway engineering.

Such a response emerges from complex micro-mechanical

processes that give rise to a highly non-linear hydro-

mechanical behaviour at the macro-scale, featuring

irreversible deformation, hysteresis, pore pressure build-

up, etc. (di Prisco & Muir Wood, 2012). The engineering

analysis of these phenomena proves even more challenging

for long-lasting cyclic loading events (‘high-cyclic’ loading),

such as those experienced by soils and foundations under

operating oﬀshore structures (e.g. oﬀshore drilling rigs,

pipelines, wind turbines) (Andersen, 2009, 2015; Randolph

& Gourvenec, 2011). A typical example is given at present

by monopile foundations for oﬀshore wind turbines, whose

design must assure full functionality of the structure during

its whole operational life – 108-109loading cycles with

alternating sequences of small-amplitude vibrations and

severe storm loading (LeBlanc et al., 2010; Abadie, 2015).

Despite the current ferment around oﬀshore wind

geotechnics (Pisanò & Gavin, 2017), frustrating uncertain-

ties still aﬀect the engineering analyses performed to assess

the capacity, serviceability and fatigue resistance of wind

turbine foundations. In this context, a major role is played

by the phenomenon of ‘sand ratcheting’: this term denotes

the gradual accumulation of plastic strains under many

loading cycles, as opposed to the occurrence of ‘shakedown’

(long-term response with no plastic strain accumulation)

(Houlsby et al., 2017). While micromechanical studies aim

to describe the occurrence and modes of sand ratcheting at

the granular level (Alonso-Marroquin & Herrmann, 2004;

Manuscript received. . .

∗Geo-Engineering Section, Faculty of Civil Engineering and

Geoscience, Delft University of Technology, Stevinweg 1, 2628

CN Delft (The Netherlands)

†Facultad de Ingeniería y Ciencias Aplicadas, Universidad de los

Andes, Mons. Aĺvaro del Portillo 12.455, 762000111, Las Condes,

Santiago (Chile)

‡Department of Civil Engineering, Faculty of Engineering,

University of Bristol, Queen’s Building, University Walk, Clifton

BS8 1TR, Bristol (United Kingdom).

McNamara et al., 2008; O’Sullivan & Cui, 2009; Calvetti

& di Prisco, 2010), usable engineering methods are cur-

rently being devised for predictions at the soil-foundation-

structure scale. Serious challenges arise in this area for at

least two reasons: (i) the time-domain, step-by-step analysis

of high-cyclic soil-structure interaction (‘implicit analysis’,

in the terminology of Niemunis et al. (2005)) is computa-

tionally prohibitive and challenging accuracy-wise; (ii) even

with viable implicit computations (e.g. through intensive

parallel computing), the literature still lacks constitutive

models reproducing cyclic sand ratcheting with satisfactory

accuracy.

To mitigate the above diﬃculties, alternative ‘explicit’

methods have been proposed, including some recent

applications to oﬀshore wind turbine foundations (Suiker

& de Borst, 2003; Niemunis et al., 2005; Achmus et al.,

2009; Wichtmann et al., 2010; Andresen et al., 2010;

Pasten et al., 2013; Jostad et al., 2014, 2015; Triantafyllidis

et al., 2016; Chong, 2017). In this framework, sand

cyclic straining is directly linked to the number of

loading cycles N– hence the term ‘explicit’. Accordingly,

the relationship between accumulated strains and N

emerges from empirical relationships accounting for micro-

structural/mechanical properties (void ratio, grain size

distribution, shear strength, etc.) and loading parameters

(stress or strain amplitude, conﬁning pressure, deviatoric

obliquity, etc.), to be calibrated based on rare high-cyclic

laboratory tests – see e.g. Lekarp et al. (2000); Suiker

et al. (2005); Wichtmann (2005); Wichtmann et al. (2005);

Wichtmann & Triantafyllidis (2015); Wichtmann et al.

(2015); Escribano et al. (2018). Most often, explicit high-

cyclic methods are used in combination with implicit

calculation stages: the latter provide the space distribution

of cyclic stress/strain increments via the time-domain

simulation of one/two loading cycles; the former feed

such information to empirical strain accumulation models

and derive global high-cyclic deformations at increasing

N(Niemunis et al., 2005; Andresen et al., 2010; Pasten

et al., 2013). Although signiﬁcantly faster than implicit time

marching, stability and accuracy issues may be experienced

in explicit N−stepping (Pasten et al., 2013).

The present work tackles the modelling of sand ratcheting

within the phenomenological framework of bounding surface

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2MODELLING THE CYCLIC RATCHETING OF SANDS

plasticity. For this purpose, the critical state SANISAND04

model by Dafalias & Manzari (2004) is enriched with a

third locus – termed ‘memory surface’ – to reproduce fabric

eﬀects relevant to cyclic ratcheting. The suitability of the

memory surface approach has been recently shown by Corti

(2016) and Corti et al. (2016) in combination with the

bounding surface model by Gajo & Muir Wood (1999b,a).

The resulting model has been successfully applied to the

cyclic analysis of certain oﬀshore soil-structure interaction

problems, involving e.g. mudmat foundations (Corti et al.,

2017) and plate anchors (Chow et al., 2015). In this work,

the SANISAND conceptual platform has been preferred

also in light of the several open-source implementations

already available (Mazzoni et al., 2007; Gudehus et al., 2008;

Ghofrani & Arduino, 2017), which will enable to move faster

towards the analysis of relevant boundary value problems.

With main focus on drained loading conditions,

the model described in the following improves the

achievements of Corti et al. in several respects: (i) general

multiaxial formulation, with pressure-sensitive hardening

rules suitable to accommodate the diﬀerent backbone

model (SANISAND04) under monotonic and cyclic loading;

(ii) improved analytical “workability” achieved through a

formulation based on the “true” stress tensor, and the use

of a memory locus with circular deviatoric section; (iii)

modiﬁed plastic ﬂow rule to reﬁne the simulation of volume

changes under cyclic loading conditions. Importantly, the

ratcheting performance of the model has been tested up

to 104loading cycles, and validated against a wider set of

literature results, including triaxial (both standard and non-

standard), simple shear and oedometer high-cyclic tests.

The ultimate goal of this paper is to help bridge

implicit and explicit approaches through the proposed

plasticity model. Its ‘implicit’ use will enable more accurate

time-domain simulations of cyclic/dynamic soil-structure

interaction under relatively short-lasting loading (e.g.

storms, earthquakes, etc.) (Corciulo et al., 2017). As more

experimental data become available and further calibration

exercises carried out, it will also contribute to explicit high-

cyclic procedures by supporting the prediction of strain

accumulation trends with lower demand of laboratory test

results.

TOWARDS A SANISAND MODEL WITH

RATCHETING CONTROL

While massive eﬀorts have been devoted to modelling the

undrained cyclic behaviour of sands, the cyclic performance

under drained loading conditions has received far less

attention. A few works tackled this issue by enhancing the

bounding surface framework with fabric-related modelling

concepts, such as Khalili et al. (2005); Kan et al. (2013);

Gao & Zhao (2015); outside traditional bounding surface

plasticity, the contributions by Wan & Guo (2001);

Di Benedetto et al. (2014); Liu et al. (2014); Tasiopoulou

& Gerolymos (2016) are also worth citing. However, none

of the mentioned works focused explicitly on drained strain

accumulation under a large number of loading cycles.

The cyclic sand model proposed in this study builds

upon two main pillars, namely the SANISAND04 model

by Dafalias & Manzari (2004) and the addition of a

memory locus accounting for fabric eﬀects during cyclic

loading (Corti, 2016; Corti et al., 2016). Since its ﬁrst

introduction in 1997 (Manzari & Dafalias, 1997), the

family of SANISAND models has expanded with new

members improving certain limitations of the original

formulation, regarding e.g. dilatancy and fabric eﬀects,

hysteretic small-strain behaviour, response to radial stress

paths, incremental non-linearity (Papadimitriou et al., 2001;

Papadimitriou & Bouckovalas, 2002; Dafalias & Manzari,

2004; Taiebat & Dafalias, 2008; Loukidis & Salgado, 2009;

Pisanò & Jeremić, 2014; Dafalias & Taiebat, 2016). In

particular, the SANISAND04 formulation includes a fabric-

related tensor improving the phenomenological simulation

of post-dilation fabric changes upon load reversals, with

beneﬁcial impact on the prediction of pore pressure build-up

during undrained cyclic loading.

Unfortunately, the set of modelling ingradients as con-

jugated in SANISAND04 cannot quantitatively reproduce

high-cyclic ratcheting, nor its dependence on relevant load-

ing parameters (especially stress obliquity, symmetry and

amplitude of the loading programme). In SANISAND04, (i)

the use of the (phenomenological) fabric tensor zis only

suitable to capture the eﬀects of initial inherent anisotropy,

as explained in detail by Li & Dafalias (2011); (ii) fabric

evolution is solely activated for denser-than-critical condi-

tions, after the stress path crosses the phase transformation

line. This latter strategy has proven not suﬃcient to capture

fabric eﬀects occurring during (drained) cyclic loading, for

instance related to the evolving distributions of voids and

particle contacts (Oda et al., 1985; O’Sullivan et al., 2008;

Zhao & Guo, 2013). A signiﬁcant impact of these facts

on numerical simulations is that the SANISAND04 model

produces only slight soil stiﬀening under drained (high-

)cyclic shear loading, resulting in exaggerated strain accu-

mulation. While acknowledging the beneﬁts of improved

fabric tensor formulations (Papadimitriou & Bouckovalas,

2002), a diﬀerent path based on the memory surface concept

will be followed in the remainder of this work.

The plasticity modelling of ratcheting phenomena has

received a few valuable contributions (di Prisco & Mortara,

2013), originally regarding metals and alloys. These

contributions have been reviewed by Houlsby et al. (2017),

and generalised into a hyper-plastic multi-surface framework

for the macro-element analysis of oﬀshore monopiles.

The present paper proposes an alternative approach

based on bounding surface plasticity and the use of an

additional memory surface to keep track of fabric changes

relevant to the ratcheting response. The concept of memory

surface (or history surface) was ﬁrst proposed by Stallebrass

& Taylor (1997) for overconsolidated clays, then applied to

sands within diﬀerent modelling frameworks by Jafarzadeh

et al. (2008); Maleki et al. (2009); Di Benedetto et al. (2014).

Herein, the latest version by Corti et al. (2016) and Corti

(2016) is adopted and enhanced within the SANISAND

family. Accordingly, the regions of the stress-space that

have already experienced cyclic loading are represented

by an evolving memory locus, within which cyclic strain

accumulation occurs at a lower rate than under virgin

loading conditions.

MODEL FORMULATION

This section presents the main analytical features of

the proposed model, with focus on embedding the

memory surface concept into the SANISAND04 backbone

formulation. Similarly to SANISAND04, the new model

is based on a bounding surface, kinematic hardening

formulation to capture cyclic, rate-independent behaviour.

The model links to the well-established Critical State theory

through the notion of ‘state parameter’ (Been & Jeﬀeries,

1985), which enables to span the behaviour of a given sand

over the loose-to-dense range with a single set of parameters.

Overall, the new model uses three relevant loci – yield,

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LIU, ABELL, DIAMBRA, PISANÒ 3

bounding and memory surface (Figure 1). All constitutive

equations are presented by ﬁrst summarising the features

inherited from Dafalias & Manzari (2004), then focusing on

the latest memory surface developments.

θ

nrMrb

θ

θ

rb

θ+π

r1

r2r3

Yield surface

Memory surface

Bounding surface

r

α

αM

θ

Fig. 1. Relevant loci/tensors in the normalised πplane.

Notation Tensor quantities are denoted by bold-face

characters in a direct notation. The symbols :, tr and hi

stand for tensor inner product, trace operator and Macauley

brackets, respectively.

σ

σ

σand ε

ε

εdenote eﬀective stress∗and strain tensor. Usual

decompositions into deviatoric and isotropic components

are used throughout, namely σ

σ

σ=s

s

s+pI

I

I(s

s

s– deviatoric

stress tensor, p= (trσ

σ

σ)/3– isotropic mean stress) and

ε

ε

ε=e

e

e+ (εvol/3)I

I

I(e

e

e– deviatoric strain tensor, εvol =trε

ε

ε–

volumetric strain). I

I

Iis the second-order identity tensor, the

deviatoric stress ratio r

r

r=s

s

s/p is also widely employed in

the formulation. The superscripts eand pare used with the

meaning of ‘elastic’ and ‘plastic’.

Model features from SANISAND04

For the sake of brevity, a multi-axial formulation is

directly provided, while conceptual discussions in a simpler

triaxial environment may be found in the aforementioned

publications. For the same reason, model details shared with

SANISAND04 are only brieﬂy recalled, whereas Table 1

provides a synopsis of all equations and material parameters

(Dafalias & Manzari, 2004).

Similarly to most SANISAND formulations, the proposed

model relies on the assumption that plastic stains only occur

upon variations in stress ratio r

r

r, so that all plastic loci

and hardening mechanisms can be eﬀectively described in

the normalised πplane (Figure 1). Importantly, the overall

formulation remains based on ‘true’ stress ratio variables,

while Gajo & Muir Wood (1999a,b); Corti et al. (2016)

used stress normalised with respect to the current state

parameter.

Elastic relationship Sand behaviour is assumed to be

(hypo)elastic inside the yield locus, with constant Poisson

ratio νand pressure-dependent shear modulus deﬁned as

∗As this work focuses on drained tests/simulations, the notation σ

σ

σ

(instead of usual σ

σ

σ0) is used for the eﬀective stress tensor with no

ambiguity.

per Richart et al. (1970); Li & Dafalias (2000):

G=G0patm[(2.97 −e)2/(1 + e)]pp/patm (1)

in which patm is the reference atmospheric pressure, G0a

dimensionless shear stiﬀness parameter, and ethe current

void ratio.

Yield locus An open conical yield locus f= 0 is used,

whose axis rotation and (constant) small opening are

governed by the evolution of the back-stress ratio α

α

αand

the parameter m:

f=p(s

s

s−pα

α

α) : (s

s

s−pα

α

α)−p2/3mp = 0 (2)

Critical state locus A unique critical state locus is

assumed and deﬁned in the multidimensional e−σ

σ

σspace.

Its projection on the e−pplane, i.e. the critical state line,

reads as (the subscript cstands for ‘critical’):

ec=e0−λc(pc/patm)ξ(3)

and requires the identiﬁcation of three material parameters

–e0,λcand ξ(Li & Wang, 1998). The aforementioned state

parameter Ψ(e, p) = e−ecquantiﬁes the distance between

current and critical void ratios (Been & Jeﬀeries, 1985; Muir

Wood & Belkheir, 1994), which is key to modelling sand

behaviour at varying relative density.

The projection of the critical state locus on the

normalised πplane can be conveniently expressed as a

deviatoric tensor r

r

rc

θ:

r

r

rc

θ=p2/3g(θ)Mn

n

n(4)

providing the critical state stress ratio associated with the

current stress ratio r

r

rthrough the unit tensor, normal to the

yield locus (Figure 1):

n

n

n= (r

r

r−α

α

α)/p2/3m(5)

The function gdescribes the Argyris-type shape of the

critical locus depending on the ‘relative’ Lode angle θ†(see

Table 1 and Dafalias & Manzari (2004)). The parameter

Mappears in its traditional meaning of critical stress ratio

under triaxial compression (directly related to the constant-

volume friction angle).

It should also be recalled that the assumption of unique

critical state locus is still a matter of scientiﬁc debate,

and certainly not the only option available – nonetheless,

a several theoretical studies may be cited in its support (Li

& Dafalias, 2011; Zhao & Guo, 2013; Gao & Zhao, 2015). An

evolving version of the locus (Equation(3)) could be adopted

in the future upon conclusive consensus on the subject – for

instance, according to the path followed by Papadimitriou

et al. (2005).

Plastic ﬂow rule Plastic strain increments – deviatoric

and volumetric – are obtained as:

de

e

ep=hLiR

R

R0dεp

vol =hLiD(6)

where R

R

R0and Dare the tensor of deviatoric plastic ﬂow

direction (Dafalias & Manzari, 2004) and the dilatancy

coeﬃcient, respectively. The plastic multiplier L(or loading

†cos 3θ=√6trn

n

n3(Manzari & Dafalias, 1997)

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4MODELLING THE CYCLIC RATCHETING OF SANDS

index) results from the enforcement of plastic consistency

and can be expressed in the following form:

L=1

Kp

∂f

∂σ

σ

σ:dσ

σ

σ(7)

with Kpcommonly referred to as plastic modulus.

Kinematic hardening and bounding surface The

back-stress ratio α

α

α(axis of the yield locus) is assumed to

evolve according to the following hardening law:

dα

α

α=2

3hLih(r

r

rb

θ−r

r

r)⇒Kp=2

3ph(r

r

rb

θ−r

r

r) : n

n

n(8)

which in turn implies the expression of Kpreported beside

(Dafalias & Manzari, 2004). According to Equation (8), the

centre of the yield locus translates in the πplane along the

r

r

rb

θ−r

r

rdirection, with magnitude governed by the hardening

factor h.r

r

rb

θrepresents the projection of the current stress

ratio onto the so-called bounding surface:

r

r

rb

θ=p2/3g(θ)Mexp(−nbΨ)n

n

n(9)

The size of the bounding surface is modulated by the

state parameter Ψand the corresponding material constant

nb. At critical state Ψ = 0 and the bounding surface

coincides with the critical locus. It is worth noting that,

for better compatibility with memory surface developments,

the present formulation reappraises projection rules based

on the stress ratio r

r

rrather than the back-stress ratio α

α

α–

compare e.g. Dafalias (1986) to Manzari & Dafalias (1997).

Additional memory surface for ratcheting control

Novel developments related to the memory surface concept

are detailed in this subsection, with direct impact on the

factors hand Din Equations (8) and (6).

Meaning and deﬁnition

Figure 1 illustrates in the normalised πplane the three main

loci involved in the model formulation:

–yield surface, distinguishes stress states associated

with either negligible or signiﬁcant plastic straining;

–memory surface, distinguishes stress states associated

with either vanishing or severe changes in granular

fabric;

–bounding surface, separates admissible/pre-failure

and ultimate stress states;

Although the above transitions may not be as sharp in

nature, the above idealisation provides conceptual input to

phenomenological constitutive modelling.

The memory locus is deployed to track the global

(re)orientation of particle contacts, and in turn the degree of

loading-induced anisotropy. Accordingly, it will be possible

to describe weak fabric changes induced by moderate high-

cyclic loads, possibly ‘overwritten’ by more severe loading

afterwards – henceforth termed ‘virgin loading’ (Nemat-

Nasser, 2000; Jafarzadeh et al., 2008).

From an analytical standpoint, the memory locus fM= 0

is represented by an additional conical surface:

fM=qs

s

s−pα

α

αM:s

s

s−pα

α

αM−p2/3mMp= 0 (10)

endowed with its own (memory) back-stress ratio and

opening variable α

α

αMand mM. As shown in the following,

the choice of a conical memory locus with circular deviatoric

section results in simpler projection rules and evolution

laws (no lengthy algebra from the diﬀerentiation of the

third stress invariant). Nevertheless, keeping the typical

Argyris-shape for the outer bounding surface (Equation (9))

preserves a dependence of both stiﬀness and strength on the

Lode angle θ.

It is postulated that, during plastic straining, (i) the

stress point on the yield surface can never lies outside

the memory surface, (ii) the memory surface can only be

larger than the elastic domain, or at most coincident. These

requisites are compatible with the following reformulation

of the hardening coeﬃcient hin Kp:

h=b0

(r

r

r−r

r

rin) : n

n

nexp "µ0p

patm n=0.5bM

bref w=2#

(11)

in which

bM= (r

r

rM−r

r

r) : n

n

n

bref = (r

r

rb

θ−r

r

rb

θ+π) : n

n

n(12)

and r

r

rb

θ+πis the opposite projection onto the bounding

surface, along the direction −n

n

nwith relative Lode angle

θ+π(Equation (9), Figure 1 – therefore, bref >0always).

The SANISAND04 deﬁnition of the hardening factor b0is

recalled in Table 1. The above deﬁnitions include the image

stress point r

r

rMon the memory surface, pointed by the unit

tensor n

n

ndeﬁned above (Equation (5)):

r

r

rM=α

α

αM+p2/3mMn

n

n(13)

The left factor in Equation (11) coincides with the

hcoeﬃcient in Dafalias & Manzari (2004) (with b0

model parameter and r

r

rin load-reversal tensor‡), whilst the

right factor introduces the memory surface concept into

SANISAND04 with the additional material parameter µ0

(Corti et al., 2016, 2017). In essence, hreceives additional

inﬂuence from the yield-to-memory surface distance bM:

as a consequence, higher Kpand soil stiﬀness result at

increasing distance bM(see evolution laws later on), but

a virgin SANISAND04 response is recovered when the yield

and the memory loci are tangent at the current stress point

σ

σ

σ≡σ

σ

σM(→bM= 0).

The two material parameters, nand w, have been pre-

set in Equation (11) to mitigate calibration eﬀorts. In

particular, extensive comparisons to experimental data (see

next sections) conﬁrmed the need for a pressure-dependent

memory surface term (Corti et al., 2017), along with

a quadratic dependence on the distance bM. Additional

experimental evidence may support in the future more

ﬂexibility about nand w, as well as other fundamental

dependences (for instance on the void ratio e).

The following subsections introduce the evolution laws

for the size and position of the memory surface, as well its

eﬀect on sand dilatancy.

Memory surface size

The expansion of the memory surface (isotropic hardening)

aims to capture phenomenologically the experimental link

between gradual change in fabric and sand stiﬀening.§This

evidence is translated into an increasing size mMof the

‡r

r

rin is the value of r

r

rat the onset of load reversal. It is updated to

current r

r

reach time the condition (r

r

r−r

r

rin) : n

n

n < 0is fulﬁlled.

§The eﬀects of a varying void ratio are already accounted for as

inheritance from SANISAND04.

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LIU, ABELL, DIAMBRA, PISANÒ 5

Table 1. Model synopsis: constitutive equations and material parameters.

Constitutive equations Material parameters

Elastic moduli G=G0patm[(2.97 −e)2/(1 + e)](p/patm )1/2G0dimensionless shear modulus

K= 2(1 + ν)G/[3(1 −2ν)] νPoisson’s ratio

Critical state line ec=e0−λc(pc/patm)ξe0reference critical void ratio

λc,ξcritical state line shape parameters

Yield function f=p(s

s

s−pα

α

α) : (s

s

s−pα

α

α)−p2/3pm m yield locus opening parameter

Memory function fM=qs

s

s−pα

α

αM:s

s

s−pα

α

αM−p2/3pmM

Deviatoric plastic ﬂow de

e

ep=hLiR

R

R0

R

R

R0=Bn

n

n−Cn

n

n2−(1/3)I

I

I

n

n

n= (r

r

r−α

α

α)/p2/3m

B= 1 + 3(1 −c)/(2c)g(θ) cos 3θ

C= 3p3/2(1 −c)g(θ)/c

g(θ) = 2c/[(1 + c)−(1 −c) cos 3θ]

Volumetric plastic ﬂow dεp

vol =hLiD

D=hA0exp βD˜

bM

dE/bref i(r

r

rd

θ−r

r

r) : n

n

nA0‘intrinsic’ dilatancy parameter

βdilatancy memory parameter

r

r

rd

θ=p2/3g(θ)Mexp(ndΨ)n

n

n ndvoid ratio dependence parameter

˜

bM

d= (˜

r

r

rd

θ−˜

r

r

rM) : n

n

n

bref = (r

r

rb

θ−r

r

rb

θ+π) : n

n

n

r

r

rb

θ+π=p2/3g(θ+π)Mexp(−nbΨ)(−n

n

n)

Yield surface evolution dα

α

α= (2/3) hLih(r

r

rb

θ−r

r

r)

r

r

rb

θ=p2/3g(θ)Mexp(−nbΨ)]n

n

n

Mcritical stress ratio (triaxial compression)

nbvoid ratio dependence parameter

ccompression-to-extension strength ratio

h=b0

(r

r

r−r

r

rin) : n

n

nexp "µ0p

patm 0.5bM

bref 2#µ0ratcheting parameter

b0=G0h0(1 −che)/p(p/patm)h0,chhardening parameters

bM= (r

r

rM−r

r

r) : n

n

n

Memory surface evolution dmM=p3/2dα

α

αM:n

n

n−(mM/ζ)fshr −dεp

volζmemory surface shrinkage parameter

dα

α

αM= (2/3) DLMEhM(r

r

rb

θ−r

r

rM)

hM=1

2"b0

(r

r

rM−r

r

rin) : n

n

n+r3

2

mMfshr h−Di

ζ(r

r

rb

θ−r

r

rM) : n

n

n#

memory surface and a larger distance between r

r

rand r

r

rMin

Equations (11)–(12). As clariﬁed in the following, variations

in size and position of the memory surface cannot be

independent, but it is convenient to address the former

aspect prior to the latter. For this purpose, the evolution

of mMis established on a geometrical basis starting from

a situation of incipient virgin loading – memory surface

coincident or tangent to the yield locus (Figure 2).

Speciﬁcally, plastic loading starting from σ

σ

σ≡σ

σ

σMis

assumed to produce a uniform expansion of the memory

surface around the pivot stress point r

r

rM

A, diametrically

opposite to r

r

rMand kept ﬁxed throughout the process. From

an analytical standpoint, this coincides with enforcing the

incremental nullity of the memory function fMat the ﬁxed

stress point A (i.e. dσ

σ

σM

A= 0):

dfMσ

σ

σM

A=∂f M

∂σ

σ

σM

A

:dσ

σ

σM

A+∂f M

∂α

α

αM:dα

α

αM+∂f M

∂mMdmM= 0

(14)

rM=r

rb

r1

r2r3

αM

α

n

θ

θ

Bounding surface

Memory surface

after expansion

Coinciding memory

and yield surfaces

before expansion

Translated yield

surface

α

rM

A

θ

Fig. 2. Memory surface expansion during virgin loading.

Trivial manipulations (see Appendix I) lead to the following

relationship:

dmM=r3

2dα

α

αM:n

n

n(15)

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6MODELLING THE CYCLIC RATCHETING OF SANDS

which is signiﬁcantly simpler than what obtained by Corti

(2016) for a memory surface with non-circular π-section.

It is further assumed by analogy that Equation (15)

determines the relationship between expansion (dmM) and

translation (dα

α

αM) of the memory locus under any loading

conditions, not only virgin.

While dmM>0(expansion) underlies ‘fabric reinforce-

ment’ and sand stiﬀening within the expanded memory

locus, an opposite eﬀect is usually induced by dilative

deformation stages and increase in void ratio (Nemat-Nasser

& Tobita, 1982). Such a ‘damage’ to the fabric conﬁguration

results in lower sand stiﬀness. Here, the suggestion by Corti

(2016) is followed, and an additional contraction term is

cast into Equation (15) to let the memory surface shrink

only during dilation (negative dεvol):

dmM=r3

2dα

α

αM:n

n

n−mM

ζfshr −dεp

vol(16)

in which the contraction term on the right is proportional

to the current locus size mMand plastic volumetric

strain increment dεp

vol, with a purely geometrical factor

fshr described with more detail in the Appendix I. The

contraction rate during dilation is governed by the material

parameter ζ, assumed for simplicity not to depend on any

stress/state variables (e.g. p,e, etc.).

Memory surface translation

In analogy with the translation rule for the yield locus, the

centre of the memory surface is assumed to translate along

the direction of r

r

rb

θ−r

r

rM(Figure 1):

dα

α

αM=2

3DLMEhM(r

r

rb

θ−r

r

rM)(17)

The hardening law (17) shares the same structure with

Equation (8), and requires a method to derive the ‘memory-

counterparts’ of the plastic multiplier and the hardening

coeﬃcient, namely LMand hM. The same approach

used for the isotropic memory hardening is re-adopted:

the translation rule for α

α

αMis rigorously speciﬁed for

virgin loading and then extended to any other conditions.

Accordingly, analytical derivations and material parameters

are substantially reduced in a way proven successful by the

the results in the following.

It is assumed that during virgin loading (σ

σ

σ≡σ

σ

σM) the

same magnitude of the incremental plastic strain can be

derived by using the yield or memory loci indiﬀerently. The

equalities below follow directly (see relevant derivations in

the Appendix I):

LM=L

hM=1

2"b0

(r

r

rM−r

r

rin) : n

n

n+r3

2

mMfshr h−Di

ζ(r

r

rb

θ−r

r

rM) : n

n

n#(18)

and are then extended by analogy to non-virgin loading.

Memory surface: eﬀect on the sand dilation

As a phenomenological recorder of fabric eﬀects, the

memory surface is also exploited to enhance the dilatancy

factor Din Equation (6), in a new way diﬀerent from

SANISAND04. The goal is to use the memory surface to

obtain increased dilatancy (or pore pressure build-up in

undrained conditions) upon load reversals following dilative

deformation (Dafalias & Manzari, 2004). For this purpose,

the memory surface is handled in combination with the

same dilatancy locus deﬁned by Dafalias & Manzari (2004),

responsible for the transition from contractive to dilative

response:

n

αM

α

r

rd

r M

rM

Memory surface

Yield surfaceYield surface

r d

Dilatancy surface

θ

Fig. 3. Geometrical deﬁnitions for the enhancement of

the dilatancy coeﬃcient.

r

r

rd

θ=p2/3g(θ)Mexp(ndΨ)n

n

n(19)

where the positive parameter ndgoverns its evolution

towards critical state (Ψ = 0). For the sake of clarity, Figure

3 displays certain geometrical quantities associated with the

relative position of the memory and dilatancy surfaces. The

distance ˜

bM

dis ﬁrst deﬁned as:

˜

bM

d= (˜

r

r

rd−˜

r

r

rM) : n

n

n(20)

with ˜

r

r

rMand ˜

r

r

rdprojections of r

r

ron the memory and

dilatancy surfaces along the −n

n

ndirection. When ˜

bM

d>0

the post-dilation contractancy produced by Din Equation

(6) is enhanced as follows:

D=hA0exp βD˜

bM

dE/bref i(r

r

rd

θ−r

r

r) : n

n

n(21)

where A0and βare two material parameters. In Equation

(21) the exponential term is deactivated by ˜

bM

d<0,

that is when the image stress ratio ˜

r

r

rMlies outside the

dilatancy surface (i.e. after dilative deformation prior to

load reversal). Conversely, additional contractancy arises

in the opposite case ˜

bM

d>0with ˜

r

r

rMlying inside the

memory surface. Compared to SANISAND04, the dilatancy

coeﬃcient accounts for fabric eﬀects through the same

memory locus employed to enhance the plastic modulus

coeﬃcient in Equation (11).

CALIBRATION OF CONSTITUTIVE PARAMETERS

The new model requires the calibration of sixteen

constitutive parameters, only one more than SANISAND04.

Two subsets parameters may be distinguished: the ﬁrst

includes material parameters already present in the original

SANISAND04 formulation – namely, from G0to nd

in Table 2; the remaining parameters govern directly

the (high-)cyclic performance under both drained and

undrained loading. The calibration of material parameters

is discussed hereafter with reference to the monotonic

and cyclic laboratory tests performed by Wichtmann

(2005) on a quartz sand – D50 = 0.55 mm, D10 = 0.29

mm, Cu=D60/D10 = 1.8(non-uniformity index), emax =

0.874,emin = 0.577. Numerical simulations are executed

with yield and memory surfaces initially coincident.

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Table 2. Model parameters for the quartz sand tested by Wichtmann (2005)

Elasticity Critical state Yield surface Plastic modulus Dilatancy Memory surface

G0ν M c λce0ξ m h0chnbA0ndµ0ζ β

110 0.05 1.27 0.712 0.049 0.845 0.27 0.01 5.95 1.01 2.0 1.06 1.17 260 0.0005 1

The calibration of the ﬁrst subset against monotonic tests

is based on the procedure detailed in Dafalias & Manzari

(2004). The shear modulus G0can be derived from the

small-strain branch of experimental stress-strain curves, or

alternatively from well-established empirical relationships

(e.g. Richart et al. (1970); Hardin & Black (1966)). A

Poisson’s ratio equal to 0.05 was assumed following the

suggestion of Dafalias & Manzari (2004) for an open-wedge

yield surface. Opening of yield surface m= 0.01 is also

consistent with the SANISAND04 model. The parameters

governing the shape of critical state line in the e−lnpplane

(e0,λcand ξ) and the critical state shear strength (Mand c)

have been identiﬁed by ﬁtting both strength and volumetric

strain trends at ultimate conditions for diﬀerent void ratios

and stress levels, as illustrated in Figure 4 by means of

deviatoric stress – axial strain (q−εa) and volumetric strain

– axial strain (εvol −εa) plots. More details about the

calibration of the remaining plastic modulus (h0,chand

nb) and dilatancy (A0and nd) parameters are available

in Dafalias & Manzari (2004) and Taiebat & Dafalias

(2008). Due to the limited availability of monotonic tests

for the considered quartz sand, these parameters have been

determined by ﬁtting the available stress – strain (q−εa)

and volumetric strain – axial strain (εvol −εa) trends

as shown in Figure 4. All calibrated soil parameters are

reported in Table 2.

The new parameters linked to the proposed memory

surface (µ0,ζand β) can be identiﬁed by best-ﬁtting cyclic

test results, possibly from both drained and undrained

triaxial cyclic tests. Here, only the drained triaxial cyclic

tests documented in Wichtmann (2005) are exploited

for calibration purposes, while their impact on the

undrained response is qualitatively discussed. In particular,

Wichtmann’s experiments concern one-way asymmetric

cyclic loading performed in two stages (Figure 5): after

the initial isotropic consolidation up to p=pin,p-constant

shearing is ﬁrst performed to reach the target average

stress ratio ηave =qave /pin; then, cyclic axial loading at

constant radial stress is applied to obtain cyclic variations

in deviatoric stress qabout the average value qave , i.e.

q=qave ±qampl (Figure 5b). High-cyclic sand parameters

are tuned to match the evolution during regular cycles of

the accumulated total strain norm εacc deﬁned as:

εacc =q(εacc

a)2+ 2 (εacc

r)2=r1

3εacc

vol 2+3

2εacc

q2

(22)

where εacc

a,εacc

r,εacc

qand εacc

vol stand for axial, radial,

deviatoric and volumetric accumulated strain, respectively.

As illustrated in Figure 6, the ratcheting response of the

soil under drained loading is governed by the µ0parameter

in Equation (11). Figure 6a proves the superior capability of

the memory surface formulation to reproduce the transition

from ratcheting to shakedown. The gradual sand stiﬀening

occurs in combination with reduced plastic dissipation, as

denoted by the decreasing area enclosed by subsequent

stress-strain loops. The sensitivity of εacc to µ0is visualised

in Figure 6b and exploited to reproduce the experimental

data from Wichtmann (2005). µ0is in this case set to 260

by ﬁtting the trend of εacc against number of loading cycles.

Dilative deformation tend to ‘damage’ the granular fabric

and thus erase ‘sand memory’. This granular process is

phenomenologically reproduced by the shrinkage of the

memory surface, at a rate governed by the parameter ζin

Equation (16). However, the eﬀect of ζ– only relevant to

stress paths beyond the dilative threshold Equation (19) – is

most apparent under undrained conditions: larger ζvalues

reduce the contraction rate of the memory surface and

postpone the build-up of positive pore pressure in the post-

dilation unloading regime (Figure 7a). Under drained high-

cyclic loading, increasing ζstill promote the aforementioned

memory surface contraction, and aﬀect soil ratcheting in the

dilative regime. For the quartz sand tested by Wichtmann

(2005), a drained high-cyclic triaxial test with stress path

crossing the phase transformation line is selected for the

calibration of the memory surface shrinkage parameter ζ.

Inﬂuence of ζon the accumulation of the total strain εacc

in Equation (22) is presented in Figure 7b. ζ= 0.0005 has

been selected to reproduce the results of high-cyclic drained

tests mobilising sand dilation, – see Figure 7b.

The last parameter βappears in the new deﬁnition of

the dilatancy coeﬃcient Din Equation (21), and mainly

controls the post-dilation reduction of the mean eﬀective

stress in undrained tests. Larger βvalues allow for larger

reductions in eﬀective mean pressure, possibly up to full

liquefaction (Figure 8a). Since the considered set of drained

test results does not support the calibration of β,β= 1

has been set judiciously with negligible inﬂuence on the

strain accumulation predicted during drained cyclic tests

(see Figure 8b). Although beyond the scope of this work on

drained strain accumulation, the marked inﬂuence of βon

the undrained response is brieﬂy illustrated in Appendix II.

MODEL PREDICTIONS OF DRAINED RATCHETING

UNDER DIFFERENT LOADING PATHS

This section overviews the predictive capability of the model

against drained high-cyclic test results from the literature.

The parameter set in Table 2 is used to simulate sand

ratcheting under diﬀerent cyclic loading conditions, namely

triaxial, simple shear and oedometer. All model results

have been obtained via single-element FE simulations

performed on the OpenSees simulation platform (Mazzoni

et al., 2007). The new model with ratcheting control has

been implemented starting from the existing SANISAND04

implementation developed at the University of Washington

(Ghofrani & Arduino, 2017).

Cyclic triaxial tests

This section considers triaxial test results from Wichtmann

(2005), not previously used for parameter calibration. The

experimental data concern the same quartz sand and both

standard and non-standard triaxial loading.

Standard triaxial loading

The model is ﬁrst validated against standard triaxial tests

of the kind sketched in Figure 5, i.e. with constant radial

stress during axial cyclic loading. The drained ratcheting

response is predicted at varying pin,ein,ηav e and qampl.

Importantly, a large number of cycles N= 104is considered,

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8MODELLING THE CYCLIC RATCHETING OF SANDS

0 5 10 15 20

0

250

500

750

1000

axial strain εa [%]

deviatoric stress q [kPa]

ein=0.59

ein=0.69

ein=0.80

0 5 10 15 20 25

−7.5

−5

−2.5

0

2.5

axial strain εa [%]

volumetric strain εvol [%]

(a) constant pin =200 kPa, varying ein

0 5 10 15 20 25

0

250

500

750

1000

axial strain εa [%]

deviatoric stress q [kPa]

pin=200 kPa

pin=100 kPa

pin=50 kPa

0 5 10 15 20 25

−7.5

−5

−2.5

0

2.5

axial strain εa [%]

volumetric strain εv [%]

(b) constant ein= 0.69, varying pin

Fig. 4. Calibration of model parameters against the monotonic drained triaxial test results by Wichtmann (2005) –

experimental data denoted by markers.

p

q

qave 1

ηave=qave/pin

pin

qampl

(a) stress path

q

Regular cycles

Number of cycles N

(b) ‘sawtooth’ cyclic loading sequence

Fig. 5. Stress paths and shear loading sequence in the tests considered for simulation (Wichtmann, 2005).

0 0.25 0.5 0.75 1

0

50

100

150

200

axial strain εa [%]

deviator stress q [kPa]

(a) Deviatoric stress- axial strain response predicted

by the new model with µ0= 100

10010 110 2103

0

0.5

1

1.5

2

number of cycles N [−]

exp (Witchmann, 2005)

µ0=60

µ0=260

µ0=560

accumulated total strain εacc [%]

(b) Inﬂuence of µ0on the accumulated total strain

norm

Fig. 6. Inﬂuence of µ0(Equation (11)) on sand response. The comparison to the experimental data by Wichtmann

(2005) refers to the following test/simulation settings: ein = 0.702, qampl = 60 kPa, pin = 200 kPa, ηave = 0.75.

and a very satisfactory agreement with experimental data

is obtained in most cases.

Inﬂuence of initial conﬁning pressure pin The

experimental data by Wichtmann (2005) show a quite

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0 500 1000 1500 2000 2500 3000 3500 4000

0

1000

2000

3000

4000

5000

6000

7000

deviatoric stress q [kPa]

mean eective stress p [kPa]

ζ=0.0005

ζ=0.0001

ζ=0.00001

(a) Post-dilation undrained unloading response

10010 110 2103

0

0.5

1

1.5

2

2.5

3

number of cycles N [−]

accumulated strain εacc [%]

exp(Wichtmann, 2005)

ζ=0.0005

ζ=0.0001

ζ=0.00001

(b) Drained high-cyclic strain accumulation

Fig. 7. Inﬂuence of ζ(Equation (16)) on sand response. Simulation settings: (a) pin=500 kPa, ein = 0.6, load reversal

at εa= 0.07; (b) pin = 200 kPa, ηave=1.125, ein = 0.68, qampl = 60 kPa.

0 500 1000 1500 2000 2500 3000 3500 4000

0

1000

2000

3000

4000

5000

6000

7000

deviatoric stress q [kPa]

mean eective stress p [kPa]

β=10

β=1

β=0

(a) Post-dilation undrained unloading response

10010 110 2103

0

0.5

1

1.5

2

2.5

number of cycles N [-]

accumulated total strain εacc [%]

β=10

β=1

β=0

(b) Drained high-cyclic strain accumulation

Fig. 8. Inﬂuence of β(Equation (21)) on sand response.

Simulation settings: (a) the quartz sand, pin=500 kPa,

ein = 0.6, load reversal at εa= 0.07; (b) the quartz sand,

pin=200 kPa, ηave =1.125, ein = 0.68, qampl = 60 kPa.

low inﬂuence of pin on the εacc −Ncurves, especially

for N < 104(Figure 9a). This is clearly in contrast with

what the new model predicts if no pressure-dependence is

incorporated in the hardening coeﬃcient h(Equation (11)),

i.e. if the exponent nof the p/patm factor is set to zero

(Figure 9b). Conversely, the intrinsic pressure-dependence

of SANISAND models can be counterbalanced through a

pressure factor (p/patm)nin h. To avoid a burst in the

number of free parameters, a default exponent n= 1/2is

adopted, also in agreement with the pressure-dependence

typically found for sand stiﬀness. The comparison between

Figures 9a and 9c proves the quantitative suitability of

Equation (11).

Inﬂuence of initial void ratio ein The experimental

evidence from Wichtmann (2005) conﬁrms the intuitive

expectation of higher strain accumulation at increasing ein

(looser sand specimens). Figure 10 illustrates the potential

10010 110 210310 4

0

0.25

0.5

0.75

1

1.25

number of cycles N [−]

accumulated total strain εacc [%]

pin=300 kPa

pin=200 kPa

pin=50 kPa

(a) Experimental data (Wichtmann, 2005)

10010 110 210310 4

0

0.25

0.5

0.75

1

1.25

number of cycles N [−]

accumulated total strain εacc [%]

pin=300 kPa

pin=200 kPa

pin=50 kPa

(b) Model simulations: n= 0

10010 110 210310 4

0

0.25

0.5

0.75

1

1.25

number of cycles N [−]

accumulated total strain εacc [%]

pin=300 kPa

pin=200 kPa

pin=50 kPa

(c) Model simulations: n= 1/2

Fig. 9. Inﬂuence of the initial mean pressure pin on

cyclic strain accumulation. Test/simulation settings:

ein=0.684, ηave =0.75, stress amplitude ratio ςampl =

qampl/pin =0.3.

of the new model to capture void ratio eﬀects, though

with slight overestimation of εacc for very dense and

very loose specimens. It is worth recalling, however, that

the parameters in Table 2 have been calibrated in the

remarkable eﬀort to capture relevant response features with

a single set of parameters.

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10 MODELLING THE CYCLIC RATCHETING OF SANDS

10010 110 210310 4

0

1

2

3

4

number of cycles N [-]

accumulated total strain εacc [%]

ein=0.803

ein=0.674

ein=0.580

(a) Experimental data (Wichtmann, 2005)

10010 110 210310 4

0

1

2

3

4

number of cycles N [−]

accumulated total strain εacc [%]

ein=0.803

ein=0.674

ein=0.580

(b) Model simulations

Fig. 10. Inﬂuence of the initial void ratio ein on cyclic strain accumulation. Test/simulation settings: pin= 200 kPa,

ηave= 0.75, qampl = 60 kPa.

Inﬂuence of cyclic stress amplitude qampl The

experimental and numerical results in Figure 11 agree on

the higher strain accumulation produced by increasing cyclic

stress amplitude qampl. In particular, satisfactory model

predictions are shown for medium-dense sand specimens

associated with ein = 0.702.

Inﬂuence of average stress ratio ηave The dependence

of sand ratcheting on the average stress obliquity

about which stress cycles occur is extremely relevant to

practical applications. Indeed, soil elements under/around a

foundation experience cyclic loading starting from diﬀerent

stress obliquities, implying diﬀerent distance from the value

related to shear failure.

Figure 12 presents another set of experimental-numerical

comparisons at varying average stress ratio ηave. The

model can reproduce the experimental increase in strain

accumulation rate for larger ηave values, although less

accurately as ηave >1. Speciﬁcally, the simulation with

ηave = 1.125 overestimates εacc signiﬁcantly when N >

1000: high-cyclic loading at large ηave jeopardises the

eﬀectiveness of the memory surface concept, as the model

tends again towards the SANISAND04 limit. While near-

failure high-cyclic loading seems not too relevant to

operational conditions in the ﬁeld, some concerns could

also be raised about the reliability of test measurements

performed under such conditions, may be due to strain

localisation phenomena (Escribano et al., 2018).

Non-standard triaxial loading

Alternative triaxial loading conditions can be generated

by varying both axial and radial stresses during the test.

As discussed by Wichtmann (2005), this can produce

‘polarised’ stress-strain cyclic paths, which seem to enhance

the tendency to strain accumulation. Unlike most modelling

exercises, the model performance is here assessed also in

relation to polarised triaxial loading. For this purpose, the

following polarisation angle αP Q and amplitude are ﬁrst

deﬁned in the Q−Pplane (Figure 13) for direct comparison

with Wichtmann’s data:

tan αP Q =Qampl

Pampl

Sampl =p(Pampl)2+ (Qampl )2

(23)

where Q=p2/3qand P=√3pare isomorphic transfor-

mations of the stress invariants pand q, and the superscript

ampl denotes cyclic variations about the initial values pave ≡

pin and qave.

The model response to non-standard triaxial loading is

compared to Wichtmann et al.’s experimental data in Figure

14 at varying polarisation angle αP Q, and Figure 15 at

varying loading amplitude Sampl. The results in Figure

14 span polarisation angles in the range from 0◦to 90◦,

and show very satisfactory εacc −Ntrends in most cases.

The only exception is the case αP Q = 10◦, in which the

model underpredicts the corresponding strain accumulation.

This singular outcome is directly caused by the analytical

expression (2) of the yield locus, conical and open-ended: in

fact, triaxial stress paths at αP Q = 10◦happen to be mostly

oriented along the uncapped zone of the elastic domain,

resulting in underestimated plastic strains.

The eﬀect of the cyclic stress amplitude Sampl at ﬁnite

polarisation angle (αP Q = 75◦) can be observed in Figure

15. Strain accumulation is accelerated by increasing Sampl,

as testiﬁed by simulation results in good agreement with all

laboratory data.

Cyclic simple shear tests

Simple shear tests are also well-established in the geo-

experimental practice, and allow to explore the soil response

to loading paths implying rotation of the principal stress

axes. Simple shear loading closely represents conditions

relevant to many soil sliding problems, e.g. in the triggering

of landslides or in the mobilisation of the shaft capacity of

piles.

The experimental work of Wichtmann (2005) also

included high-cyclic simple shear tests on the same quartz

sand previously tested in the triaxial apparatus – the

validity of the same sand parameters in Table 2 can

be thus assumed. Two types of cyclic simple shear tests

were performed: (i) cyclic shear loading applied along a

single direction; (ii) so-called cyclic multidimensional simple

shear (CMDSS) tests, in which the direction of shear

loading is shifted by 90◦in the horizontal plane after, in

this case, N= 1000 cycles. As all tests were performed

under controlled shear strain amplitude, the experimental

results were visualised in terms of residual (plastic) strain

accumulation – following the deﬁnition (22), the residual

strain in strain-controlled simple shear tests coincides with

the permanent vertical strain.

Experimental and numerical curves corresponding

with cyclic shear strain amplitude γampl = 5.8×10−3

are compared in Figure 16, where the dashed lines

denote the shift in shear loading direction at N=

1000 – relevant to CMDSS tests. Despite unavoidable

stress/strain inhomogeneities in simple shear experiments

(Dounias & Potts, 1993), reasonably similar residual strain

accumulations are displayed in Figures 16a–16b. The

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10010 110 210310 4

0

0.5

1

1.5

2

number of cycles N [-]

qampl=80 kPa

qampl=60 kPa

qampl=31 kPa

accumulated total strain εacc [%]

(a) Experimental data (Wichtmann, 2005)

10010 110 210310 4

0

0.5

1

1.5

2

number of cycles N [−]

qampl=80 kPa

qampl=60 kPa

qampl=31 kPa

accumulated total strain εacc [%]

(b) Model simulations

Fig. 11. Inﬂuence of the cyclic stress amplitude qampl on cyclic strain accumulation. Test/simulation settings: pin=

200 kPa, ηave= 0.75, ein = 0.702.

10010 110 210310 4

0

0.5

1

1.5

2

2.5

number of cycles N [−]

ηave=1.125

ηave=1

ηave=0.75

ηave=0.375

accumulated total strain εacc [%]

(a) Experimental data (Wichtmann, 2005)

10010 110 210310 4

0

0.5

1

1.5

2

2.5

η=1.125

ηave=1

ηave=0.75

ηave=0.375

accumulated total strain εacc [%]

number of cycles N [−]

(b) Model simulations

Fig. 12. Inﬂuence of the average stress ratio ηave on cyclic strain accumulation. Test/simulation settings: pin= 200

kPa, ein= 0.684, qampl= 60 kPa.

P

Q

Pampl

Qampl

αPQ αPQ= 0º

αPQ= 90º

√2/3 qave

√3 pave

Sampl

Fig. 13. Non-standard triaxial stress paths in the Q−P

plane as deﬁned by Wichtmann (2005).

model is also able to capture the temporary increase in

accumulation rate produced by the sudden change in shear

loading direction.

Cyclic oedometer tests

Cyclic oedometer test results are more rare in the literature,

nonetheless a recent instance is reported by Chong &

Santamarina (2016) for three diﬀerent sands (a blasting

sand, Ottawa F110 and Ottawa 50–70). The following

simulations regard oedometer tests on Ottawa 50–70

specimens (D10 = 0.26 mm, D50 = 0.33 mm, Cu= 1.43,

emax = 0.87,emin = 0.55) prepared at two diﬀerent void

ratios, ein = 0.765 and ein = 0.645. Both loose and dense

specimens were subjected to stages of monotonic-cyclic-

monotonic loading, at either low or high vertical static stress

(Figure 17a): (i) low static load – monotonic compression

up to 100 kPa →cyclic vertical loading in the range 200–

100 kPa (100 cycles) →monotonic re-compression up to

1.4 MPa; (ii) high static load: monotonic compression up

to 1 MPa →cyclic vertical loading in the range 1.1–1

MPa (100 cycles) →monotonic re-compression up to 1.4

MPa. Regarding the data set in Chong & Santamarina

(2016), a slightly diﬀerent calibration approach had to be

followed: ﬁrst, the thirteen SANISAND04 parameters (from

G0to nd) have been identiﬁed based on drained monotonic

triaxial tests on Ottawa sand from the literature (Lin et al.,

2015); then, the oedometer high-cyclic response has been

simulated by either (i) keeping the same (µ0, ζ , β)set in

Table (Figure 17b), or adjusting the three parameters for

best-ﬁt purposes (Figure 17c). Experimental and numerical

results are compared in Figure 17 in terms of void ratio vs

vertical stress curves, for consistency with the original plots

in Chong & Santamarina (2016).

As apparent in Figure 17, most experimental-numerical

mismatch is produced during monotonic loading stages,

which a model with an open conical yield surface is

not suited to reproduce. As for cyclic compaction, the

parameters calibrated from Wichtmann’s tests tend to

underpredict the reduction in void ratio. In contrast,

satisfactory numerical results are displayed in Figure 17c

after re-calibrating µ0and ζas well (Table 3, same β).

There are two steps in re-calibrating µ0and ζ: (1) the loose

specimen is selected to calibrate µ0parameter under the

loading condition that cyclic vertical loading in the range

200–100 kPa, since under this condition the parameters ζ

and βhave no impact on the cyclic behaviour; (2) the dense

sample under the same cyclic loading conditions is selected

Prepared using GeotechAuth.cls

12 MODELLING THE CYCLIC RATCHETING OF SANDS

100101102103

0

0.5

1

1.5

number of cycles N [-]

αPQ=0°

αPQ=10°

αPQ=30°

αPQ=54.7°

αPQ=75°

αPQ=90°

accumulated total strain εacc [%]

(a) Experimental data (Wichtmann, 2005)

100101102103

0

0.5

1

1.5

number of cycles N [-]

αPQ=0°

αPQ=10°

αPQ=30°

αPQ=54.7°

αPQ=75°

αPQ=90°

accumulated total strain εacc [%]

(b) Model simulations

Fig. 14. Inﬂuence of the polarisation angle αPQ on cyclic strain accumulation. Test/simulation settings: pin= 200

kPa, ein= 0.69, ηave = 0.5, stress amplitude in the Q−Pplane Sampl= 60 kPa.

10010 110210 3

0

0.5

1

1.5

Sampl=20 kPa

Sampl=40 kPa

Sampl=60 kPa

Sampl=80 kPa

number of cycles N [-]

accumulated total strain εacc [%]

(a) Experimental data (Wichtmann, 2005)

10010 110 2103

0

0.5

1

1.5

number of cycles N [-]

accumulated total strain εacc [%]

Sampl=20 kPa

Sampl=40 kPa

Sampl=60 kPa

Sampl=80 kPa

(b) Model simulations

Fig. 15. Inﬂuence of the stress amplitude Sampl in the Q−Pplane on cyclic strain accumulation. Test/simulation

settings: pin= 200 kPa, ein= 0.69, ηav e= 0.5, αP Q = 75◦.

0 500 1000 1500 2000

0

2.5

5

7.5

number of cycles N [−]

residual strain εacc

res [%]

standard simple shear

CMDSS

(a) Experimental results

0 500 1000 1500 2000

0

2.5

5

7.5

number of cycles N [−]

standard simple shear

CMDSS

residual strain εacc

res [%]

(b) Simulation results

Fig. 16. Cyclic simple shear tests (single loading direction and CMDSS) – comparison between experimental results

and model predictions. Test/simulation settings: σa= 24 kPa (initial vertical stress), ein= 0.69, γampl = 5.8×10−3.

Table 3. Model parameters for the Ottawa sand tested by Chong & Santamarina (2016)

Elasticity Critical state Yield surface Plastic modulus Dilatancy Memory surface

G0ν M c λce0ξ m h0chnbA0ndµ0ζ β

90 0.05 1.28 0.8 0.012 0.898 0.7 0.01 5.25 1.01 1.2 0.4 1.35 44 0.005 1

to calibrate ζwith the µ0determined in step (1). Other

simulations are conducted with the same parameters. The

model captures two expected, yet relevant, aspects:

– at given initial vertical stress, the looser sand compact

more than the dense sand; for a given initial void

ratio, higher initial compression level results in lower

cyclic compaction;

– after cyclic loading, the void ratio evolves during re-

compression towards the initial virgin compression

line (Figure 17c).

The results in Figure 17 conﬁrm the remarkable predictive

potential of new model. It could also be shown that fully

realistic values of the horizontal-to-vertical stress ratio are

obtained, owing to the rotational mechanism of the narrow

yield surface (and regardless of the low Poisson’s ratio

selected – Table 3). The predictions of the monotonic

oedometer response could be improved by introducing a

capped yield surface as proposed by Taiebat & Dafalias

(2008).

Prepared using GeotechAuth.cls

LIU, ABELL, DIAMBRA, PISANÒ 13

0 500 1000 1500

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

void ratio e

vertical stress σv [kPa]

∆e=0.0099 ∆e=0.0027

∆e=0.0049 ∆e=0.0018

(a) Experimental data (Chong &

Santamarina, 2016)

0 500 1000 1500

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

vertical stress σv [kPa]

void ratio e

Δe=0.0018 Δe=0.0011

Δe=0.0016 Δe=0.0007

(b) Model simulations – µ0and ζ

from Table 2

0 500 1000 1500

0.6

0.62

0.64

0.66

0.68

0.7

0.72

0.74

0.76

0.78

0.8

vertical stress σv [kPa]

void ratio e

Δe=0.0098 Δe=0.0026

Δe=0.0053 Δe=0.0018

(c) Model simulations – all param-

eters from Table 3

Fig. 17. Cyclic oedometer tests – comparison between experimental results and model predictions.

COMPLIANCE WITH THE CONCEPT OF

‘TERMINAL DENSITY’

While experimental and numerical results were compared

above in terms of strain norm εacc (Equation (22)), it is also

interesting to inspect the accumulation of volumetric (εacc

vol )

and deviatoric (εacc

q) strains individually – as exempliﬁed in

Figure 18. Based on experimental observations, Wichtmann

(2005) concluded that the εacc

vol /εacc

qratio mainly depends

on the average stress ratio ηave held during cyclic loading.

Other factors like void ratio, conﬁning pressure and

stress amplitude seemed to play limited roles. The new

model is found to reproduce such a ratio correctly in

the medium/high strain range, although with an overall

underestimation of εacc

vol (Figure 18 – note that the

experimental and predicted trend lines become parallel for

εacc

q>0.4).

0 0.1 0.2 0.3 0.4 0.5

0

0.1

0.2

0.3

0.4

0.5

accumulated deviatoric strain εq

acc [%]

accumulated volumetric strain εvol

acc [%]

model simulation

exp (Wichtmann, 2005)

Fig. 18. High-cyclic evolution of deviatoric and volumet-

ric strain under drained triaxial loading. Test/simulation

settings: pin= 200 kPa, ein = 0.7, ηave = 0.75, qampl =60

kPa, N= 104.

It is believed that these inaccuracies relate mostly to the

assumed modelling of sand dilatancy, future eﬀorts will be

spent to remedy this shortcoming. However, it is also worth

reﬂecting here on the link between Wichtmann’s results and

other related published results. In particular, Narsilio &

Santamarina (2008) postulated on an experimental basis the

existence of a so-called ‘terminal density’, that is a state of

constant void ratio and steady fabric – including critical

state as a particular instance. Every sand appears to attain

a speciﬁc terminal density depending on initial, boundary

and loading conditions (Narsilio & Santamarina, 2008;

Chong & Santamarina, 2016), with direct inﬂuence on the

observed accumulation of all strain components. However,

the experimental trend in Figure 18 from Wichtmann (2005)

does not seem to evolve towards such a terminal state.

Further studies about such a discrepancy and, more widely,

about the existence and the properties of terminal density

loci will positively aﬀect future modelling eﬀorts on the

high-cyclic response of soils.

CONCLUSIONS

The critical state, bounding surface SANISAND04 model

was endowed with an additional locus in the stress space

(memory surface) to improve the simulation of high-cyclic

sand ratcheting under a variety of initial, boundary and

loading conditions. The constitutive equations, directly

presented in a multi-axial framework, were implemented

in the ﬁnite element code OpenSees, based on an existing,

open-source implementation of SANISAND04. Compared to

previous formulations, the proposed models proved more

reliable in capturing the dependence of sand ratcheting,

as well as potentially more ﬂexible in terms of mean

eﬀective pressure decay under undrained loading. Extensive

validation against experimental results was performed with

regard to triaxial (standard and non-standard), simple shear

and oedometer drainded cyclic tests.

The impact of this and future work on the subject will

link to further calibration eﬀorts against new high-cyclic

datasets, still rare in the scientiﬁc literature and usually

out of the scope of industry projects. It is anticipated

that deeper insight and more reliable empirical correlations

Prepared using GeotechAuth.cls

14 MODELLING THE CYCLIC RATCHETING OF SANDS

may be obtained for a range of sandy materials. This will

support the use of ratcheting models, both implicit and

explicit, in the (likely) lack of speciﬁc evidence about strain

accumulation trends.

NOTATIONS

A0‘intrinsic’ dilatancy parameter

˜

bM

drelative position of the memory and the

dilatancy surfaces

bMyield-to-memory surface distance

b0hardening factor

bref reference distance for normalisation

Cuuniformity coeﬃcient

ccompression-to-extension strength ratio

chhardening parameter

Ddilatancy coeﬃcient

D10,50,60 the diameter in the particle-size distribu-

tion curve corresponding to 10%,50% and

60% ﬁner

e

e

edeviatoric strain tensor

evoid ratio

e0reference critical void ratio in Equation 3

ecvoid ratio at critical state

ein initial void ratio

fyield function

fMmemory function

fshr memory surface shrinkage geometrical

factor

Gshear modulus

G0dimensionless shear modulus

g(θ)interpolation function for Lode angle

dependence

˜

hvirgin state hardening factor generalised

into common situations

hhardening factor

hMmemory-counterpart of the hardening

coeﬃcient

h0hardening parameter

I

I

Isecond-order identity tensor

Kpplastic modulus

KM

pmemory-counterpart of the plastic modu-

lus

Lplastic multiplier

LMmemory-counterpart of the plastic multi-

plier

Mcritical stress ratio in compression

myield locus opening parameter

mMmemory locus opening parameter

Nnumber of loading cycles

n

n

nunit tensor normal to the yield locus

n

n

nMunit tensor for memory surface contraction

npre-set material parameter

nb,d void ratio dependence parameters

Pisomorphic transformation of the stress

invariant p

Pampl cyclic amplitude of P

pmean eﬀective stress

patm atmospheric pressure

pin initial eﬀective mean stress

Qisomorphic transformation of the stress

invariant q

Qampl cyclic amplitude of Q

qdeviatoric stress

qampl cyclic deviatoric stress amplitude

qave average deviatoric stress

R0

R0

R0deviatoric plastic ﬂow direction tensor

R

R

Rplastic strain rate direction tensor

r

r

rdeviatoric stress ratio tensor

r

r

rMimage deviatoric stress ratio point on the

memory locus

r

r

rb

θ+πprojection onto the bounding surface with

relative Lode angle θ+π

r

r

rb,c,d

θbounding, critical and dilatancy deviatoric

stress ratio tensor

r

r

rC,D projection of r

r

ralong −n

n

nMon memory

surfaces after and before contraction

r

r

rin initial load-reversal tensor

˜

r

r

rprojection of r

r

ron the yield surface along

−n

n

n

˜

r

r

rMprojection of r

r

ron the memory surface

along −n

n

n

˜

r

r

rdprojection of r

r

ron the dilatancy surface

along −n

n

n

Sampl cyclic polarisation stress amplitude

s

s

sdeviatoric stress tensor

wpre-set material parameter

x1,2,3line-segments deﬁned to derive memory

surface contraction law

αP Q polarisation angle

α

α

αback-stress ratio tensor

α

α

αMmemory back-stress ratio tensor

βdilatancy memory parameter

ε

ε

εstrain tensor

εacc accumulated total strain

εacc

a,r,q,vol accumulated axial, radial, deviatoric and

volumetric strain

εa,vol axial and volumetric strains

εp

vol plastic volumetric strain

ηave average deviatoric stress ratio in triaxial

space

γampl cyclic shear strain amplitude

λcCSL shape parameter

µ0ratcheting parameter

νPoisson’s ratio

Ψstate parameter

σ

σ

σstress tensor

σ

σ

σMimage stress tensor on the memory locus

σ

σ

σM

Astress tensor at point A

ςampl cyclic stress amplitude ratio

θrelative Lode angle

ξCSL shape parameter

ζmemory surface shrinkage parameter

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APPENDIX I – ANALYTICAL DERIVATIONS

This appendix collects relevant analytical derivations,

skipped in the main text for better readability.

Memory surface expansion

The expansion law for the memory surface is derived

through the consistency condition in Equation (14) applied

to the stress point A (Figure 2). Importantly, the partial

derivative ∂f M/∂α

α

αMat point Acan be expressed as follows:

∂f M

∂α

α

αMA

=−pr

r

rM

A−α

α

αM

q(r

r

rM

A−α

α

αM) : (r

r

rM

A−α

α

αM)

=pn

n

n(A1)

due to the position of Ataken diametrically opposed to

the projection of the stress point on the memory surface

(see Figure 2, along −n

n

n). After computing the derivative

∂f M/∂ mMand setting dσ

σ

σM

A= 0, the evolution law (15)

results from Equation (14).

Memory surface contraction

The geometrical factor fshr in Equation 16 is evaluated to

prevent the memory surface from shrinking smaller than the

elastic domain.

It is assumed that the shrinkage of the memory surface

occurs at ﬁxed image stress ratio r

r

rM, along the direction of

the unit tensor n

n

nM:

n

n

nM=r

r

rM−r

r

r

p(r

r

rM−r

r

r) : (r

r

rM−r

r

r)(A2)

The following segments along the n

n

nMdirections are

deﬁned in agreement with in Figure A1:

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LIU, ABELL, DIAMBRA, PISANÒ 17

n

αMα

rM

nM

r

rD

rC

r M

r2r3

r3

Dilatancy surface

Memory surface

before contraction

Memory surface

after contraction

Yield surface

Fig. A1. Geometrical contraction mechanism of the

memory surface.

x1=n

n

nM: (r

r

rM−r

r

r)

x2=n

n

nM: (r

r

r−r

r

rD) = n

n

nM: (r

r

r−˜

r

r

r)

x3=n

n

nM: (r

r

rM−r

r

rC) = n

n

nM: (r

r

rM−˜

r

r

rM)

(A3)

where

˜

r

r

r=α

α

α−p2/3mn

n

n˜

r

r

rM=α

α

αM−p2/3mMn

n

n(A4)

It should be recalled that, during virgin loading, r

r

rM=r

r

rand

n

n

nM=n

n

n. To avoid undesired intersections, the contraction

of the memory surface is gradually decelerated as the yield-

memory tangency is approached. For this purpose, the

factor fshr in Equation 16 is deﬁned as per Corti (2016):

fshr = 1 −x1+x2

x3

(A5)

Under general conditions, points Cand Ddo not

coincide, so that the segment x3is longer than segment

x1+x2. Therefore, (x1+x2)/x3<1⇒fshr >0and the

contraction mechanism is carried on until the segment CD

vanishes.

Memory surface translation

As postulated in the relevant section, it assumed that

during virgin loading (σ

σ

σ≡σ

σ

σM) the same magnitude of the

incremental plastic strain can be derived by using the yield

or memory loci indiﬀerently:

kdε

ε

εpk

kR

R

Rk=L=1

Kp

∂f

∂σ

σ

σ:dσ

σ

σ=1

KM

p

∂f M

∂σ

σ

σM:dσ

σ

σM=LM

(A6)

where R

R

R=R

R

R0+DI

I

I/3(see Equation (6)). Under virgin

loading conditions, LM=Land KM

p=Kphold rigorously.

By enforcing plastic consistency on the memory surface,

it can be found that:

∂f M

∂σ

σ

σM:dσ

σ

σM=−∂f M

∂α

α

αM:dα

α

αM+∂f M

∂mMdmM=LMKM

p

(A7)

After setting L=LMand substituting the partial

derivatives of the memory function fMwith respect to α

α

αM

and mM, Equation A7 can be rewritten as:

LKM

p=pn

n

n:dα

α

αM+r2

3pdmM(A8)

Introducing Equation (17) into the above equation leads to:

KM

p=2

3p"hM(r

r

rb

θ−r

r

rM) : n

n

n+r3

2

1

LdmM#(A9)

Imposing virgin loading (r

r

r=r

r

rM) into Equation (8) yields:

Kp=2

3ph(r

r

rb

θ−r

r

r) : n

n

n=2

3p˜

h(r

r

rb

θ−r

r

rM) : n

n

n(A10)

with

˜

h=b0

(r

r

rM−r

r

rin) : n

n

n(A11)

Under virgin loading conditions, KM

p=Kp. Combining

Equation A9 with A10 results in:

˜

h(r

r

rb

θ−r

r

rM) : n

n

n=hM(r

r

rb

θ−r

r

rM) : n

n

n+r3

2

1

LdmM(A12)

The combination of Equations (6), (16) (A11) and (A12)

leads to the ﬁnal Equation (18).

APPENDIX II – IMPACT ON UNDRAINED CYCLIC

RESPONSE

While the main body of the paper focused on the modelling

and simulation of drained cyclic strain accumulation, some

space is given in this appendix to compare the proposed

model and the parent SANISAND04 formulation in terms

of undrained cyclic performance. For this purpose, the

experimental results from Ishihara et al. (1975) are taken as

a reference after Dafalias & Manzari (2004) – in particular,

an undrained cyclic triaxial test performed on a Toyoura

sand specimen at constant cell pressure pin = 294 kPa,

initial void ratio ein = 0.808 and amplitude of applied

deviatoric stress qampl = 114.2kPa. The SANISAND04

parameters shared by the proposed model are reported in

TableA1 as identiﬁed by Dafalias & Manzari (2004).

The comparison between experimental data and

SANISAND04 simulation is reported in Figure A2a.

The SANISAND04 model captures the cyclic decrease in

eﬀective mean stress, in a way positively aﬀected by the

enhanced post-dilation contractancy achieved through the

fabric-tensor formulation. However, the model does not

accurately predict the pore pressure build-up during each

cycle and, in turn, the number of cycles required to reach

the phase transformation line (PTL). Conversely, Figure

A2b shows the improved performance of the proposed

model, based on the memory surface concept combined

with the new dilatancy coeﬃcient deﬁned in Equation (21).

The proposed model accounts for the gradual stiﬀening

over consecutive cycles and predicts better the number

of loading cycles to phase transformation. Comforting

predictions are also obtained in terms of pore pressure

vaules beyond phase transformation, along with the nearly

nil eﬀective mean stress reached upon unloading. Overall,

the new memory-surface-based ﬂow rule seems a promising

alternative to the approach followed by Dafalias & Manzari

(2004).

As highlighted in Figure A2c in comparison to Figure

A2b, the proposed model oﬀers higher ﬂexibility in

reproducing the undrained cyclic behaviour, depending on

the set of cyclic parameters µ0,ζand βselected. In

general, the initial pore pressure build-up prior to phase

transformation can be controlled through the parameter

µ0; the post-dilation stress path is mainly governed by the

parameter β, which aﬀects indirectly the shrinkage of the

memory surface – the larger β, the smaller the minimum

eﬀective stress reached upon post-dilation unloading.

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18 MODELLING THE CYCLIC RATCHETING OF SANDS

Table A1. Toyoura sand parameters shared by SANISAND04 and the new model – after tests by Ishihara et al.

(1975)

Elasticity Critical state Yield surface Plastic modulus Dilatancy

G0ν M c λce0ξ m h0chnbA0nd

125 0.05 1.25 0.712 0.019 0.934 0.7 0.01 7.05 0.968 1.1 0.704 3.5

0 50 100 150 200 250 300

−150

−100

−50

0

50

100

150

mean eective stress p [kPa]

deviatoric stress q [kPa]

Exp (Ishihara et al. 1975)

SANISAND04 z max

=5, c

z =600

(a) Comparison between experimental result (Ishi-

hara et al., 1975) and SANISAND04 simulation

result (Dafalias & Manzari, 2004).

0 50 100 150 200 250 300

−150

−100

−50

0

50

100

150

mean eective stress p [kPa]

deviatoric stress q [kPa]

Exp (Ishihara et al. 1975)

New model µ0=45, β=16.5

(b) Comparison between experimental result (Ishi-

hara et al., 1975) and new model simulation result

(µ0= 45,ζ= 0.00001,β= 16.5).

0 50 100 150 200 250 300

−150

−100

−50

0

50

100

150

mean eective stress p [kPa]

deviator stress q [kPa]

New model µ0=140 β=0

Post-dilation stress path,

eect of β Below PTL, eect of μ0

(c) Inﬂuence of µ0and βon the undrained

performance of the new model (µ0= 150,ζ= 0.00001,

β= 0).

Fig. A2. Undrained cyclic behavior of Toyoura sand.

Test/simulation settings: pin = 294 kPa, ein = 0.808,

qampl = 114.2kPa.

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